Properties

Degree 2
Conductor $ 2^{2} \cdot 5 \cdot 401 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.62·3-s − 5-s + 4.42·7-s − 0.358·9-s − 1.81·11-s − 1.06·13-s + 1.62·15-s − 1.23·17-s + 4.12·19-s − 7.18·21-s − 4.61·23-s + 25-s + 5.45·27-s + 0.847·29-s − 0.697·31-s + 2.95·33-s − 4.42·35-s + 3.45·37-s + 1.72·39-s − 3.20·41-s − 10.5·43-s + 0.358·45-s + 4.77·47-s + 12.5·49-s + 2.00·51-s + 3.05·53-s + 1.81·55-s + ⋯
L(s)  = 1  − 0.938·3-s − 0.447·5-s + 1.67·7-s − 0.119·9-s − 0.548·11-s − 0.295·13-s + 0.419·15-s − 0.299·17-s + 0.946·19-s − 1.56·21-s − 0.962·23-s + 0.200·25-s + 1.05·27-s + 0.157·29-s − 0.125·31-s + 0.514·33-s − 0.747·35-s + 0.567·37-s + 0.276·39-s − 0.501·41-s − 1.60·43-s + 0.0533·45-s + 0.695·47-s + 1.79·49-s + 0.280·51-s + 0.420·53-s + 0.245·55-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(8020\)    =    \(2^{2} \cdot 5 \cdot 401\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{8020} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 8020,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;5,\;401\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;5,\;401\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 \)
5 \( 1 + T \)
401 \( 1 - T \)
good3 \( 1 + 1.62T + 3T^{2} \)
7 \( 1 - 4.42T + 7T^{2} \)
11 \( 1 + 1.81T + 11T^{2} \)
13 \( 1 + 1.06T + 13T^{2} \)
17 \( 1 + 1.23T + 17T^{2} \)
19 \( 1 - 4.12T + 19T^{2} \)
23 \( 1 + 4.61T + 23T^{2} \)
29 \( 1 - 0.847T + 29T^{2} \)
31 \( 1 + 0.697T + 31T^{2} \)
37 \( 1 - 3.45T + 37T^{2} \)
41 \( 1 + 3.20T + 41T^{2} \)
43 \( 1 + 10.5T + 43T^{2} \)
47 \( 1 - 4.77T + 47T^{2} \)
53 \( 1 - 3.05T + 53T^{2} \)
59 \( 1 + 10.5T + 59T^{2} \)
61 \( 1 - 9.43T + 61T^{2} \)
67 \( 1 - 4.76T + 67T^{2} \)
71 \( 1 - 0.533T + 71T^{2} \)
73 \( 1 + 7.64T + 73T^{2} \)
79 \( 1 + 9.62T + 79T^{2} \)
83 \( 1 + 6.01T + 83T^{2} \)
89 \( 1 + 6.21T + 89T^{2} \)
97 \( 1 - 12.6T + 97T^{2} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−7.48674409451086900052931446814, −6.88634946067532080342897789028, −5.87308374073477706880596139041, −5.35849556332201492360488011872, −4.79479984264219487340544391726, −4.20116805809227019825419048355, −3.08013178892788415757231656670, −2.10085883885856545653107520726, −1.13948588617764530255932568053, 0, 1.13948588617764530255932568053, 2.10085883885856545653107520726, 3.08013178892788415757231656670, 4.20116805809227019825419048355, 4.79479984264219487340544391726, 5.35849556332201492360488011872, 5.87308374073477706880596139041, 6.88634946067532080342897789028, 7.48674409451086900052931446814

Graph of the $Z$-function along the critical line